Prove that sin(180+x)=-sinx geometrically without using the expansion of sin(a+b). I have worked on it a lot but at the end every ends up up making a ray which makes an angle 'x' with the positive x-axis. Then this ray is revolved counterclockwise/anticlockwise so as to make the the angle made by the ray with the x-axis as 180+x. Then they say that as the first ray lies in the first quadrant there the sin of x is positive there because perpendicular and hypotenuse are both positive. For the second ray they say that it is symmetrically opposite to the first ray and so its base is negative, perpendicular is negative and hypotenuse is positive (this ray lies in the 3rd quadrant). So they say sin of this (180+x) is negative! But I fail to understand that how is this angle (180+x) contained in the triangle formed in the 3rd quadrant? The triangle that is made in the third quadrant by the revolved ray has an angle 'x' and not(180+x). Also as a matter of fact how can a triangle have an angle greater than 180 as its interior angle? It clearly violated the angle sum property of a triangle. That's why I need help in understanding this. Thanks